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Full text of "Treatise On Analysis Vol-Ii"

246       XIV   INTEGRATION IN LOCALLY COMPACT GROUPS

We shall now show that

(14.1.5.3)                    !(/+/') = !(/) + !(/')

for all/,/' in Jf * . By (iii), it is enough to prove that!(/) + !(/') <; !(/+/').
Let s > 0 be a real number, and let h be a function ^0 belonging to Jf R(G)
such that h(x) ^ 1 on the union of the (compact) supports off and/' ((3.18.2)
and (4.5.2)). Then it is enough to prove that there exists a compact neighbor-
hood V of e in G, such that, for all g e 3C*+ with Supp(^) c= V, we have

(14.1.5.4)              (/: g) + (/': g) ^ (/ + /': g) + e(h : g).

To prove this, put u =/+/' +ie/z, and let v (resp. z/) be the function
which coincides with//w (resp./'/w) on Supp(/+/') and is zero on the com-
plement of this set. Since at every frontier point x of Supp(/+/') we have
/(*) +/'(*) = 0> and therefore f(x) =/'(*) = 0, it follows that v and v'
belong to JfR(G) and are ^0. The functions v and v' are therefore uniformly
continuous with respect to a left-invariant distance defining the topology of G
((12.9.1) and (3.16.5)), and therefore, for each rj > 0, there exists a compact
neighborhood V of e such that \v(s) - v(t)\ < Y\ and \v'(s) - v'(t)\ s; i; for all
pairs (j, 0 such that s""1/e V. Now let getf% be such that Supp(^) <z V.
For each j e G we have t?  j(s)g g (u(j) + jy)  y(%. (This is obviously true at
points where j(s)g vanishes, hence at points outside ,sV; and if / e ^V we have
v(t) ^ v(s) + rj.) Similarly, we have u7  y(s)g  (v'(s) + ri)j(s)g. This being so,
let c (1 ^ / ri) be real numbers ^0, and st (1 g i ^ /z) elements of G such

that tf 5  ciY(si)9> Then we have

i=l

n

/ = vu   c, v             _

i=l                             =1

and a similar set of inequalities with/replaced by/'. Consequently
(/ : 0) + (/': 0) g  c(t?(s,) + i;^) + 2iy) ^ (1 4- 2^)^,.,

because u 4- u' ^ 1. From the definition of w, and properties (ii), (iii), and (v),
we obtain

(/: 0) + (f'9) g (1

^ (/+/' : g)
so that, by taking q such that

we get (14.1.5.4)all//' e ^f(G, Up) n ^f + and all , because/i-^In( |/j) is a seminorm.